Suppose there are servers $S_1,S_2,S_3$ such that they have service times following an exponential distribution with rates $mu_1,mu_2,mu_3$, respectively.

Upon entering the system a person must be serviced in the following order : $S_1 to S_2 to S_3$

Consider the scenario where once we arrive there is only one customer ahead of us at $S_3$.

I am in need of a clarification for the expected time in the system given $S_3$ is occupied when we arrive to use it (we are under the assumption if a server is busy we must wait for it to clear the current user).

Using a prepared solution we get the following:

Let $T:= $ time in system.

$displaystyle E(T mid S_3 text{ is occupied}) = bigg(frac{mu_1}{mu_1+mu_3}bigg)_{(1)}cdotbigg(frac{mu_2}{mu_1+mu_3}bigg)_{(2)}bigg(frac{1}{mu_3}bigg)_{(3)}$

I have used subscripts to label each term.

Now, $(1)$ makes sense as this is service time at $S_1$ being less than the service time at $S_3$. Additionally, $(3)$ makes sense as this is expected service time at $S_3$.

The clarification I am after is for $(2)$, it seems we would want $displaystyle frac{mu_2}{mu_2+mu_3}$ because this would be that the service time at $S_2$ is less than service time at $S_3$. However in the solution the denominator in $(2)$ is $mu_1+mu_2$.

Is the solution correct? If so can you provide an explanation as why $(2)$ has this form?